Approximate First Integrals for a System of Two Coupled Van Der Pol Oscillators with Linear Diffusive Coupling

I Naeem,F Mahomed

doi:10.3390/mca15040720

Abstract

The approximate partial Noether operators for a system of two coupled van der Pol oscillators with linear diffusive coupling are presented via a partial Lagrangian approach. The underlying system of two equations, in general, do not admit a standard Lagrangian. However, the approximate first integrals are constructed by utilization of the partial Noether’s theorem with the help of approximate partial Noether operators associated with a partial Lagrangian. These approximate partial Noether operators are not approximate symmetries of the system under study and they do not form an approximate Lie algebra. Moreover, we show how approximate first integrals can be constructed for perturbed ordinary differential equations (ODEs) without making use of a standard Lagrangian.

Highlights

The system of two coupled van der Pol oscillators with linear diffusive coupling was first proposed by Rand and Holmes [1]
Where A and B are coupling parameters that give the strength of the interaction and y and z are dependent variables which model the state of the oscillators and prime denotes the differentiation with respect to x
The first integrals for a system of coupled van der Pol oscillators (1) with the help of approximate partial Noether operators corresponding to the partial Lagrangian (2) are determined from the formula

Summary

INTRODUCTION

The system of two coupled van der Pol oscillators with linear diffusive coupling was first proposed by Rand and Holmes [1]. For the system of two weekly coupled nonlinear oscillators y′′ = −ω12y + εα12z, z′′ = −ω22z + 2εα1yz, where ω1, ω2, ε and α1 are positive constant and prime denotes the differentiation with respect to x, no variational problem exists. The theory of the approximate partial Noether operators and approximate conservation laws for differential equations with a small parameter has been recently introduced by Johnpillai et al in [7] (see [8] for ODEs). We give an easy way to construct approximate partial Noether operators and approximate first integrals for perturbed equations without making use of a Lagrangian. Such a system is known as an approximate partial Euler-Lagrange system

PARTIAL NOETHER OPERATORS

FIRST INTEGRALS

CONCLUDING REMARKS